Political maths

[Gob]

Could it be that Labour leader Ed Miliband's demand that all school pupils must study maths until they are 18 has been prompted by new evidence that his own MPs struggle with numbers?



The man in charge of the party's policy review, Jon Cruddas, admitted this weekend that he is "barely numerate". And when the Royal Statistical Society (RSS) recently tested the ability of honourable members to answer a relatively simple mathematical question, only a quarter of Labour MPs got it right.

A total of 97 MPs were asked this probability problem: if you spin a coin twice, what is the probability of getting two heads?*

Among Conservative members, 47% gave the wrong answer, which is disappointing enough. But of the 44 Labour MPs who took part, 77% answered incorrectly.

(*The correct response, of course, is 25%.)



The survey also asked MPs if they generally felt confident when dealing with numbers -

76% of Tories said they did
72% of Labour MPs surveyed expressed confidence

However, when asked if they thought politicians use official statistics and figures accurately when talking about their policies, only 17% of Conservative respondents agreed, as did 30% of the Labour members who took part.

I wish I had been a fly-on-the-wall when the Ipsos Mori pollsters conducted the survey. The maths question was put to 41 Conservative MPs, 44 Labour MPs, nine LibDems and three from other parties in face-to-face interviews.

Given the confidence in their numeracy expressed at the beginning of the survey, I wonder how the 60% of members who got the answer to the probability question wrong felt by the end.

The research was commissioned by the Getstats committee at the RSS (of which I am a member), part of a 10-year long campaign to improve the way Britain handles numbers.

[dales]

Hey, our MP's (presidents) are dumber than yours!

[MajGenl.Meade]

I must be qualified to be an MP. The answer is clearly 50%

If you flip a coin once, the chances of it being head/tails is of course 50%.

But if you flip it the first time and it's tails, then it is impossible to flip it a second time to get a SECOND heads result. You've already failed. Probability of getting two heads - zero

So in order to actually flip a coin twice and get two heads in succession, the first flip MUST have come up heads.

So now on your second flip your odds of getting heads are 50%.

[Sean]

Not quite Gen'l...

The question clearly stated that you are flipping a coin twice, the second flip is not dependant on the result of the first flip. Each flip gives a 50% chance of heads so overall, your chances are 25%.

So there!

[MajGenl.Meade]

Of course the odds rely upon the result of the first toss!

The chances of getting that first "heads" is 50%. No question about that.

BUT if the first flip came up "tails", the probability of flipping it a second time to get two heads in a row are now zero. Can't be done. No probability whatsoever. Experiment over and done with.

Let's assume that on the first toss "heads" did come up. Now what are the chances of the second flip coming up "heads"?

50% again. QED. They should have asked the probability of flipping TWO coins at the same time and both coming up heads.

Mind you, I should defer to you as you are no doubt a great tosser

Meade

[oldr_n_wsr]

Every coin toss is 50-50 chance. You can spin it any way you want, but each one is 50-50.


Now if you are talking Craps, Good looking ladies rolling dice will more often than not, make you money, which you will then spend on their services, so it is a wash (and a shower, and a rub down, not in that order)

[dgs49]

My goodness.

There are four possibilities, to wit,
HH,
HT,
TT, and
TH.

There is one chance in four that one will flip HH. Which was the question.

[MajGenl.Meade]

Let me repeat the question:

if you spin a coin twice, what is the probability of getting two heads?

If it's tails the first time, you will not and you cannot spin it a second time to get a "second heads" result. Game over. Finished. Done. No chance. 0% chance of spinning two heads.

I challenge you to actually try it - spin the coin once. Is it tails? Experiment concludes.

This is not a mathematical question but one of the English language and logic. I can't help it if question setters are illiterate morons

As I said, they should have asked the probability of spinning two coins and getting two heads at the same time.
That would be a 25% probability.

Meade

[Gob]

If it's tails the first time, you will not and you cannot spin it a second time to get a "second heads" result.

But the probability remains at 25%, the first outcome does not preclude subsequent attempts.

[MajGenl.Meade]

It said "twice" - that means twice. Not three times or 53. The first outcome precludes the second one. Why is this so difficult to understand? IT WAS A STUPIDLY WORDED QUESTION

[Sean]

I can see where you are going wrong Meade...
You are setting the probability after rather than before the first toss.

You're actually twice the tosser you think you are...

[MajGenl.Meade]

Wouldn't that be half? For some obscure reason I am reminded of Upchuck and Di. She vacated this planet due to carpool/tunnel syndrome. Something to do with aristofcracy? Dodi and the handjive?

And no, I am not (setting etc.). Probability is always conditional. The probability of any event reoccuring is (whether in a series of two or more) a variable number. The actuality of the previous event(s) affects the likelihood of subsequent ones.

Before you can spin two heads, you must spin one. So beforethe first spin, there is no "probability" at all of spinning two heads in two spins - there is only a probability of spinning one. The second is dependent upon the first. Before the first spin, the probability of achieving the essential result is 50%.

After the first spin, and before the second spin, there is a new probability of spinning two heads in a row (in just two spins - not 32). It is either 50% or 0%.

Thus the probability of achieving two heads in two spins is different to the probability for achieving it in three or four or five. The greater the number of repetitions, the more likely one is to get two heads come up (not necessarily consecutively - this example is to illustrate the change of probability in general).

(Although it may apply consecutively - still thinking about that). But I revert to the original contention - the proper way to apply a probability of 25% to achieving two heads in two spins is to use two coins. Simultaneously. The theory is correct but the application is wrong. Stupid question. Not stupid answers.

Meade

[Gob]

Sorry Meade, you're wrong. Dave nailed it above.

[Gob]

if you spin a coin twice, what is the probability of getting two heads?

Spin one: heads / spin two: heads.

Spin one: heads / spin two: tails.

Spin one: tails / spin two: heads.

Spin one: tails / spin two: tails.

One in four.

[Sean]

Before you can spin two heads, you must spin one. So beforethe first spin, there is no "probability" at all of spinning two heads in two spins - there is only a probability of spinning one. The second is dependent upon the first. Before the first spin, the probability of achieving the essential result is 50%.

This is where your logic falls down Meade. Of course there is a probability before the first spin and it is precisely that probability which was asked for in the original question.

If you were correct, then no bookie could ever give odds on Spurs conceding two goals in a game as there is no guarantee of them conceding a first goal*.






*Okay, bad example...

[Econoline]

Meade, the conditions of the experiment specify that you must toss the coin twice, not just toss it twice under certain conditions (i.e., only if the first toss is heads). Each pair of coin tosses is one iteration of the experiment, and each pair of tosses will result in one of four outcomes: HH, HT, TH, or TT. Do the experiment--the entire experiment, not just half of it--a sufficiently large number of times and 25% of the outcomes will be HH.

I believe the (related) question you might be attempting to answer is "If you toss a coin and it comes up heads, what are the odds it will come up heads again if you toss it a second time?"--which is of course 50%.

Another way to think of the problem which might help: if you asked an actuary to predict the odds of a husband and wife both surviving to the age of 90, he wouldn't be able to answer, "It depends on the age at which the first of the two dies, because if that first spouse dies at an age of less than 90 then there is no chance that both will survive to the age of 90." IOW, the second part of that reply would be true, but the reply does not answer the question which was asked.

[MajGenl.Meade]

When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities describe the statistical number of outcomes considered divided by the number of all outcomes (tossing a fair coin twice will yield HH with probability 1/4, because the four outcomes HH, HT, TH and TT are possible). When it comes to practical application, however, the word probability does not have a singular direct definition.

Conditional probability is the probability of some event A, given the occurrence of some other event B.

Granted - there is a 1 in 4 chance that a hypothetical coin tossed twice might come up HH viewed from one single perspective. But from the conditional probability view, the actual ability to spin a second H is dependent upon the first H having occurred. Assuming the first perspective, before tossing the coin the first time a possible answer in the theoretical sense is 25%. But this changes as soon as the first toss has been made - it is no longer true. And if it is no longer true after one toss why should it be "true" before any coins have been tossed? Because it is one possible answer specific to one possible point in time - which the actual experiment changes to two points in time.

Besides, how many times do I have to do this

Meade

Before anyone notices?

Meade

[oldr_n_wsr]

At least a few more time.

[Lord Jim]

Granted - there is a 1 in 4 chance that a hypothetical coin tossed twice might come up HH viewed from one single perspective.

Correct.

But from the conditional probability view

Which was not stated or implied in the question.

Besides, how many times do I have to do this

Meade

Before anyone notices?

Meade

Here's the flag you ought to be raising Gen'l:

[MajGenl.Meade]

It does not have to be stated or implied. A person is perfectly entitled to view the same question from different views of probability - theoretical and practical views are equally valid

As to the flag - the word "Meade" followed by "Bullshit" should be a fairly clear communication I would have thought.


Meade